Find all the values of 𝑏𝑏 that will make the trinomial 4𝑥^2 + 𝑏x − 6 factorable?
Choose one value of b and factor the resulting trinomial.

To determine the values of \( b \) that make the trinomial \( 4x^2 + bx - 6 \) factorable, we need to find values of \( b \) that allow it to be factored into two binomials.

The trinomial is in the form \( ax^2 + bx + c \), where \( a = 4 \), \( b = ? \), and \( c = -6 \). For this trinomial to be factorable using integers, we need to find factors of \( a \times c = 4 \times (-6) = -24 \) whose sum is equal to \( b \).

### Step 1: List factor pairs of -24
We need factor pairs of -24 that add up to \( b \):
- \( 1 \) and \( -24 \) (sum = -23)
- \( -1 \) and \( 24 \) (sum = 23)
- \( 2 \) and \( -12 \) (sum = -10)
- \( -2 \) and \( 12 \) (sum = 10)
- \( 3 \) and \( -8 \) (sum = -5)
- \( -3 \) and \( 8 \) (sum = 5)
- \( 4 \) and \( -6 \) (sum = -2)
- \( -4 \) and \( 6 \) (sum = 2)

### Step 2: Solve for \( b \)
Based on these pairs, the possible values of \( b \) are:
- \( b = -23 \)
- \( b = 23 \)
- \( b = -10 \)
- \( b = 10 \)
- \( b = -5 \)
- \( b = 5 \)
- \( b = -2 \)
- \( b = 2 \)

### Step 3: Choose one value of \( b \)
Let's choose \( b = 5 \).

### Step 4: Factor the trinomial
The trinomial is now:
\[
4x^2 + 5x - 6
\]
We look for two numbers whose product is \( 4 \times (-6) = -24 \) and whose sum is \( 5 \). These numbers are \( 8 \) and \( -3 \).

We rewrite the middle term \( 5x \) as \( 8x - 3x \), and then factor by grouping:
\[
4x^2 + 8x - 3x - 6
\]
Group the terms:
\[
(4x^2 + 8x) - (3x + 6)
\]
Factor out the greatest common factors from each group:
\[
4x(x + 2) - 3(x + 2)
\]
Now, factor out the common binomial factor \( (x + 2) \):
\[
(4x - 3)(x + 2)
\]

Thus, the factored form of \( 4x^2 + 5x - 6 \) is:
\[
(4x - 3)(x + 2)
\]

Would you like more details or have any questions about this process?

### 5 Related Questions:
1. What is the connection between the discriminant and factorability?
2. Can all quadratic expressions be factored into integers?
3. How would you solve a quadratic trinomial by completing the square?
4. What does the quadratic formula reveal about the roots of quadratics?
5. How can you verify the factorization by expanding the binomials?

### Tip:
Always check your factorization by expanding it back to the original form to ensure correctness!

Find all the values of b that will make the trinomial 4x^2 + bx − 6 factorable? Choose one value of b and factor the resulting trinomial.

Discover how to find the values of b that make the trinomial 4x^2 + bx - 6 factorable. This problem covers algebraic concepts like quadratic equations, factoring trinomials, and the zero product property. Includes a step-by-step example for b = 5, showing the complete factorization.

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